Not Applicable.
Reference to a xe2x80x9cComputer Listing Appendix Submitted on a Compact Discxe2x80x9d
A Computer Program Listing Appendix of the programming language, which can be used to practice the method of the present invention, is submitted with this application on two identical compact discs (CD). The compact discs are labeled Copy 1 and Copy 2. Copy 1 is entitled xe2x80x9c010811xe2x80x941041xe2x80x9d and copy 2 is entitled xe2x80x9c010811xe2x80x941038.xe2x80x9d Each CD is hereby incorporated herein by reference.
The CDs are write-only and are IBM-PC compatible. Each compact disc contains ASCII text files xe2x80x9cdata_feb25.mxe2x80x9d and xe2x80x9cadapt.mxe2x80x9d disclosing a computer program and parameter values, respectively, which can be used to demonstrate the method and system of the present invention. The file xe2x80x9cdata_feb25.mxe2x80x9d was created Jul. 17, 2000, and the file contains 2,202 bytes. The file xe2x80x9cadpat.mxe2x80x9d was created Feb. 27, 2000, and the file contains 137 bytes.
1. Field of the Invention
The present invention relates to a method and system for stabilizing a rotor about its geometric center in a magnetic bearing system at a constant rotor speed. In the method, the controller for controlling the magnetic bearing uses an adaptive control algorithm which simultaneously identifies and compensates for synchronous sensor runout and rotor mass unbalance while determining a control action that drives the rotor rotating at a constant speed to its geometric center. Sensor runout and mass unbalance is determined by varying the magnetic stiffness of the magnetic bearing which is achieved by perturbation of the bias currents in opposing electromagnet coils using an algorithm that does not alter the equilibrium of the rotor while the rotor is rotating at a constant speed.
2. Description of Related Art
Periodic disturbances are common in rotating machinery. Compensating for such disturbances is critical to the performance of systems using active magnetic bearings. The two dominant sources of periodic disturbances in magnetic bearings are synchronous sensor runout and mass unbalance. Mass unbalance, which results from a lack of alignment between the principal axis of inertia and the geometric axis of the rotor, generates a force disturbance synchronous with rotor angular speed. Runout originates from non-uniform electrical and magnetic properties around the sensing surface and lack of concentricity of the sensing surface. It generates a disturbance in rotor position at multiple harmonics of the frequency of rotation. Both synchronous sensor runout and mass unbalance are unavoidable since they result from manufacturing imperfections and cause rotor vibration, degrade performance, and can lead to instability if they are not adequately compensated.
Although the problem of simultaneous compensation of mass unbalance and synchronous sensor runout has appeared in the literature only recently (Kanemitsu, et al., Identification and Control of Unbalance and Sensor Runout on Rigid Active Magnetic Bearing Systems, 5th Int. Symp. on Magnetic Suspension Technol., Santa Barbara, Calif. (1999); Setiawan, et al., Adaptive Compensation of Sensor Runout and Mass Unbalance in Magnetic bearing Systems, IEEE/ASME Int. Conf. on Advanced Intelligent Mechatronics, Atlanta, Ga. (1999); Setiawan, et al., ASME J. Dyn. Sys. Meas. and Cont. 123: 211 (2001); Sortore, Observer Based Critical Response in Rotating Machinery, PhD Dissertation, University of Virginia, Charlottesville, Va. (1999)), a large volume of research exists on compensation of the individual disturbances. Some of the early work on mass unbalance compensation is based on insertion of a notch filter in the control loop (Beatty, Notch Filter Control of Magnetic Bearings, MS Thesis, Mass. Institute of Technology, Cambridge, Mass. (1988)). The drawback of this approach stems from negative phase of the notch transfer function which can reduce the stability margin of the closed-loop system and lead to instability (Bleuler, et al., IEEE Trans. on Control Sys. Tech. 2: 280-289 (1994); Na and Park, J. Sound Vibration 201: 427-435 (1997)). Another popular approach is adaptive feedforward control (Hu and Tomizuka, ASME J. Dyn. Sys. Meas., and Cont. 115: 543-546 (1993); Shafai, et al., IEEE Control Sys. 14: 4-13 (1994)), where Fourier coefficients of the disturbance are estimated and cancelled on-line. Operationally, these controllers resemble notch filters (Na and Park, ibid.) and can result in instability if designed without considering the underlying structure of the closed-loop system. To preserve stability, Herzog, et al. (IEEE Trans. on Control Sys. Technol. 4: 580-586 (1996)) developed the generalized notch filter and Na and Park (ibid.) proposed variation of the least mean square algorithm. Other approaches that compensate for mass unbalance while ensuring stability include adaptive auto-centering (Lum, et al., IEEE Trans. on Control Sys. Technol. 4: 587-597 (1996)) and output regulation with internal stability (Matsumura, et al., Modeling and Control of Magnetic Bearing Systems Achieving a Rotation Around the Axis of Inertia, 2nd Int. Symp. on Magnetic Bearings, Tokyo, Japan. pp 273-280 (1990)). Both of these approaches stabilize the rotor about its mass center.
Though mass unbalance compensation has been widely studied with the objective of stabilization about the mass center, most commercial applications require geometric centering to avoid seal wear. The problem of geometric center stabilization has been addressed by a few researchers (Hisatani and Koizumi, Adaptive Filtering for Unbalance Vibration Suppression, 4th Int. Symp. on Magnetic Bearings, ETH Zurich, Switzerland (1994); Song and Mukherjee, Integrated Adaptive Robust Control of Magnetic bearings, IEEE Int. Conf. on System, Man, and Cybernetics, Beijing, China (1996)), but more general results (Reinig and Desrochers, ASME J. Dyn. Sys. Meas. Cont. 108: 24-31 (1986); Mizuno, An Unified Approach to Controls for Unbalance Compensation in Active Magnetic Bearings, IEEE Int. Conf. on Control Applications, Italy (1998)) establish that mass or geometric center stabilization can be achieved through cancellation of the disturbance in current or displacement signal, respectively. In a general approach for disturbance attenuation, Knospe, et al., J. Vibration and Control 2: 33-52 (1996); Knospe, et al., ASME J. Dyn. Sys. Meas. Cont. 119: 243-250 (1997)) claimed that any form of vibration, which can be measured, can be attenuated using a pseudo-inverse of the pre-computed influence coefficient matrix. The performance of the algorithm amidst uncertainties was investigated and experiments used to demonstrate effectiveness. The method decouples the problem into two independent tasks, and while it has been demonstrated to work successfully, there is no theoretical basis for stability of the two interacting processes. Other approaches employed for disturbance compensation include robust control designs (Fujita, et al., Experiment on the Loop Shaping Based H-Infinity Control of Magnetic Bearing, Proc. Am. Control Conf. (1993); Rutland, et al., Comparison of Controller Designs for attenuation of Vibration in a Rotor Bearing System under Synchronous and Transient Conditions, 4th Int. Symp. on Magnetic Bearings, ETH Zurich, Switzerland, pp 107-112 (1994); Setiawan, et al., ibid. (1999)), Q-parameterization control (Mohamed et al., Q-parameterization Control of Vibrations in a Variable Speed Magnetic Bearing, IEEE Int. Conf. on Control Applications, Hartford, Conn. (1997)), and off-line adaptation (Kim and Lee, IEEE/ASME Trans. on Mechatronics 2:51-57 (1997)). Among them, the work by Kim and Lee (ibid.) and Setiawan et al., ibid. (2001) address the problem of sensor runout compensation.
Unfortunately, none of the above approaches lend themselves to mass unbalance compensation in the presence of significant synchronous sensor runout. This problem, widely acknowledged in the literature but essentially unsolved, stems from lack of observability of disturbances with the same frequency content. A credible way to distinguish between these disturbances is to perturb the operating conditions of the plant or its parameters, but recent research (Kanemitsu, et al., ibid.; Setiawan, et al., ibid. (1999); Sortore, ibid.) that proposes rotor speed variation is not acceptable for a number of applications. Therefore, there still remains a need for a way to distinguish between these disturbances at constant rotor speed. This invention presents a method for controlling magnetic bearings that can distinguish between the two disturbances and enable the rotor to be stabilized around its geometric center and thereby improve the usefulness of magnetic bearings.
The present invention provides a method and system for stabilizing a rotor about its geometric center in a magnetic bearing system at a constant rotor speed. In the method, the controller for controlling the magnetic bearing uses an adaptive control algorithm which simultaneously identifies and compensates for synchronous sensor runout and rotor mass unbalance while determining a control action that drives the rotor rotating at a constant speed to its geometric center. Sensor runout and mass unbalance is identified by varying the magnetic stiffness which is achieved by perturbation of the bias currents in opposing electromagnet coils using an algorithm that does not alter the equilibrium of the rotor while it is rotating at a constant speed.
Therefore, the present invention provides a method for simultaneous identification and compensation of sensor runout and mass unbalance of a rotor rotating at a constant speed in a magnetic bearing which is under the control of a controller for controlling the currents in the electromagnetic coils in the magnetic bearing, comprising (a) varying the magnetic stiffness of the magnetic bearing by excitation of bias currents in the electromagnet coils of the magnetic bearing using an algorithm which causes persistency of excitation for identification of synchronous periodic disturbances such as sensor runout and mass unbalance by continuously varying over time the currents to the electromagnetic coils in the magnetic bearing about their nominal values as a function of an independent time function to generate a series of excitations in the currents levitating a rotor in the magnetic bearing without disturbing the equilibrium of the rotor rotating at the constant speed; (b) identifying the sensor runout and mass unbalance for an excitation in the series of excitations in the currents using a second algorithm consisting of adaptation laws which determines values for the sensor runout and mass unbalance at the excitation; (c) compensating for the sensor runout and mass unbalance using a third algorithm that uses the values identified from step (b) to determine a control action that modifies the current levitating the rotor; and (d) repeating steps (a) to (c) until the rotor is stabilized about its geometric center as it is rotating at the constant speed in the magnetic bearing.
In a further embodiment of the above method, after the rotor has been stabilized about its geometric center, steps (a) to (d) are repeated at a regular interval to maintain the rotor about its geometric center.
The present invention further provides a method for stabilizing a rotor rotating at a constant speed about its geometric center in a magnetic bearing without disturbing the equilibrium of the rotor rotating at the constant speed comprising (a) providing a magnetic bearing including plurality of position sensor means wherein each position sensor means provides a signal as a measure of the position of the rotor in the air gap and a plurality of electromagnetic coils of the magnetic bearing; (b) providing a rotational speed sensor means for determining the speed of the rotor of the magnetic bearing; (c) providing an angular position sensor means for determining the angular position of the rotor in the magnetic bearing; (d) providing a generator means for providing currents to each of the electromagnetic coils for levitating the rotor in the magnetic bearing; and (e) providing a controller means including an adaptive control framework which over a period of time is sufficient to stabilize the rotor at its geometric center wherein the adaptive control framework uses a persistency of excitation algorithm to direct the generator means to introduce over time a series of successive excitations provided to the electromagnetic coils wherein each excitation changes the stiffness of the magnetic field of the magnetic bearing which over time generates persistency of excitation without affecting equilibrium of the rotor, which allows the controller means to simultaneously identify synchronous periodic disturbances in the rotor rotating at constant speed for each excitation measured by the position sensors using an adaptation laws algorithm and determine a control action using a control action algorithm that compensates for the synchronous periodic disturbances for the excitation measured by the position sensor which alters the currents provided to the electromagnetic coils by the generator means for levitating the rotor rotating at the constant speed and to drive the rotor to its geometric center which for each successive excitation drives the rotor closer to its geometric center until the rotor is stabilized about its geometric center.
In a further embodiment of the present invention, the periodic disturbances are synchronous sensor runout and mass unbalance.
In a further still embodiment of the method, after the rotor has been stabilized about its geometric center, the controller at regular intervals redetermines the geometric center of the rotor using the persistency of excitation algorithm, the adaptation laws algorithm, and the control action algorithm to maintain the rotor about its geometric center.
The present invention further provides a method for stabilizing over time a rotor rotating at a constant speed about its geometric center in a magnetic bearing without disturbing the equilibrium of the rotor rotating at the constant speed which comprises (a) providing a controller for the magnetic bearing that determines a current to each electromagnetic coils in the magnetic bearing to levitate the rotor about its geometric center; and (b) providing a program for the controller comprising (i) a persistency of excitation algorithm that enables the controller to continuously vary over time the currents to the electromagnetic coils in the magnetic bearing about their nominal values as a function of an independent time function which generates a series of excitations without disturbing the equilibrium of the rotor rotating at the constant speed (ii) an adaptation laws algorithm that enables the controller in response to an excitation in the series and a signal from a position sensor that provides signals corresponding to the position between the rotor and the electromagnetic coils of the magnetic bearing corresponding to the excitation in the series to determine sensor runout and mass unbalance for the rotor rotating at the constant speed; and (iii) a control action algorithm that enables the controller in response to the sensor runout and mass unbalance determined using the adaptation laws algorithm to determine a control action that modifies the currents for levitating the rotor to compensate for the sensor runout and mass unbalance determined for the excitation in the series, wherein the program enables the controller to modify the currents for levitating and stabilizing the rotor about its geometric center.
In further embodiment of the above method, the nominal values of the bias currents in the opposed paired electromagnetic coils are those that provide a force that cancels the weight of the rotor when the rotor is geometrically centered.
In an embodiment further still of the above method, after the rotor has been stabilized about its geometric center, the controller at regular intervals redetermines the geometric center of the rotor using the bias current excitation algorithm, the adaptation laws algorithm, and the control action algorithm to maintain the rotor about its geometric center.
In any one of the above embodiments of the method of the present invention, the magnetic bearing is a radial magnetic bearing or a thrust magnetic bearing.
The present invention further provides in a magnetic bearing apparatus comprising a stator assembly with a radial arrangement of a plurality of electromagnetic coils mounted in a stator assembly around a cylindrical opening in which a magnetic field is generated by each of the electromagnetic coils for levitating a rotor, the improvement which comprises: control means for stabilizing the rotor when rotating at a constant speed wherein the control means varies magnetic stiffness of the magnetic bearing by introducing excitations into the currents to the electromagnetic coils wherein the excitations to the current to one of the electromagnetic coils is proportionally related to the excitations to the current to the other electromagnetic coils which generates persistency of excitation without disturbing the equilibrium of the rotor rotating at a constant speed and which enables the control means to simultaneously identify and compensate for sensor runout and mass unbalance in determining a control action for stabilizing the rotor about its geometric center.
The present invention further provides a computer simulation model for determining the parameters for stabilizing a rotor rotating at a constant speed in a magnetic bearing apparatus, comprising (a) providing a computer program comprising (i) a persistency of excitation algorithm that continuously varies over time the currents to the electromagnetic coils in the magnetic bearing about their nominal values as a function of an independent time function which generates a series of excitations without disturbing the equilibrium of the rotor rotating at the constant speed; (ii) an adaptation laws algorithm that in response to an excitation in the series and a signal corresponding to the rotor position in the air gap between the rotor and the electromagnetic coils corresponding to the excitation in the series determines sensor runout and mass unbalance for the rotor rotating at the constant speed; and (iii) a control action algorithm that in response to the sensor runout and mass unbalance determined using the adaptation laws algorithm to determine a control action that modifies the currents for levitating the rotor to compensate for the sensor runout and mass unbalance determined for the excitation in the series, wherein the program modifies the currents for levitating the rotor and stabilizing the rotor about its geometric center.
The above computer simulation model is useful for simulation of a radial magnetic bearing or a thrust magnetic bearing.
The present invention further provides a system for simultaneous identification and compensation of sensor runout and mass unbalance of a rotor rotating at a constant speed in a magnetic bearing which is under the control of a controller for controlling the currents to the electromagnetic coils in the magnetic bearing, comprising (a) varying the magnetic stiffness of the magnetic bearing by excitation of currents in the electromagnet coils of the magnetic bearing using an algorithm which causes persistency of excitation for identification of the synchronous disturbances, such as sensor runout and unbalance, by continuously varying over time the currents to the electromagnetic coils in the magnetic bearing about their nominal values as a function of an independent time function to generate a series of excitations in the currents levitating a rotor in the magnetic bearing without disturbing the equilibrium of the rotor rotating at the constant speed; (b) identifying the sensor runout and mass unbalance for an excitation in the series of excitations in the currents using a second algorithm consisting of adaptation laws which determines values for the sensor runout and mass unbalance at the excitation; (c) compensating for the sensor runout and mass unbalance using a third algorithm that uses the values from step (b) to determine a control action that modifies the current levitating the rotor; and (d) repeating steps (a) to (c) until the rotor is stabilized about its geometric center as it is rotating at the constant speed in the magnetic bearing.
In a further embodiment of the above system, after the rotor has been stabilized about its geometric center, steps (a) to (d) are repeated at a regular interval maintain the rotor about its geometric center.
In an embodiment further still of the above system, the magnetic bearing is a radial magnetic bearing or a thrust magnetic bearing.
In any one of the above embodiments of the present invention, the algorithm for persistency of excitation nominally chooses bias currents in the opposite magnetic coils to provide a force that cancels weight of the rotor when it is geometrically centered wherein the bias currents nominally satisfy the relation
k(i210xe2x88x92i220)=m{overscore (g)}l2
wherein k is a magnetic force constant, m is the rotor mass, {overscore (g)} is acceleration due to gravity, l is nominal air gap between the rotor and electromagnetic coils, and i10 and i20 are the bias currents for the opposite electromagnetic coils, and then the excitations in the opposite electromagnetic coils are determined according to the relations
i10=i*10=xcex41, i20=i*20+xcex42
wherein i*10 and i*20 are constants and xcex41 and xcex42 are the bias current excitations which are of small magnitude and wherein to prevent rotor oscillation due to the bias current excitations, xcex41 and xcex42 are chosen according to the relation
xcex42=(i*10/i*20)xcex41, xcex41=Asin(xcfx89et), xcfx89e less than xcfx89
wherein A is the amplitude of the bias current excitation, xcfx89e is the frequency of the bias current excitation, and xcfx89 is the angular rotation of the rotor.
Therefore, it is an object of the present invention to provide a method and system for stabilizing a rotor about its geometric center in a magnetic bearing system at a constant rotor speed.
In particular, it is an object of the present invention to provide a method and system for stabilizing a rotor about its geometric center in a magnetic bearing by identifying and compensating for synchronous periodic disturbances such as sensor runout and mass unbalance while the rotor is rotating at a constant speed.
These and other objects of the present invention will become increasingly apparent with reference to the following drawings and preferred embodiments.